Alright, sorry for the late reply.
I'm not live streaming so yes I can fix it in post. when we shot the gaming stuff last year, we would take break every 10 mins or so and would a clap and point the camera at the tv to sync the gameplay footage with the camera and the audio.
We also ended up having a friend live mix the audio if we had more then 2 mics and send that mix to the camera, but it's not ideal given there's a lot of laughing out burst or a bit of shouting
So for audio we are using AT 803a wired lavs going to a presonus RM32 AI. If it's only 2 people we hard pan the mics and send it to the camera (Nikkon D3200) so we can tweek em in post ( last year we used an old ramsa mixer with dbx 166xl to limit and boost the mics)
For the gameplay footage, all the older consoles with rf output(atari, intellivision...) are all link and passing thru a vcr and that going to a composite matrix with with the rest of the consoles. That matrix is going to a no name "all to hdmi" scaller, that scaller has a few hdmi inputs so all the rest of modern consoles are switched with that. the output is then send to a 2 out HDMI DA, so one output to the tv, the other to the avermedia Extremecap U3 also knows as the CV710
We did a bunch of test going direct and skipping all the switchers and das and none of that gear seems to affect the timming, those clip are always in sync with the camera using the original software or the premier pro pluggin (wich works better) So the camera and this recording would always almost be perfectly synced for 20-30 min files.
We would always have sync problems with the seperated audio recording from the pc and we tried a few different 2 chanel audio interface, like an M-box, Tascam 180, the line in from the hdmi recorder.
So the main thing we want to do is to record multi track audio from the presonus, so we have all the mics in iso, so that we can do the best mix possible and record the hdmi source on the same pc, idealy in the same software to save time to manualy sync the camera to the audio and hdmi clips.